Appears in collection : Higher Algebra, Geometry, and Topology / Algèbre, Géométrie et Topologie Supérieures

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. To obtain it, we prove a parametrized version of the classical result, due to Kadeishvili and Stasheff, that the cohomology of a Poincaré duality space carries a cyclic C-infinity algebra structure. I will also discuss how this higher structure can be used to relate seemingly disparate problems in commutative algebra and differential topology: on one hand, the problem of putting multiplicative structures on minimal free resolutions and, on the other hand, the question of whether a given Poincaré duality fibration can be promoted to a smooth manifold bundle.